There are a lot of number theory problems like the following:
(1) Find all $m$ such that $m=a^2+b^2$, or all $m$ such that $m=a^2+b^2+c^2$, or even $m=a^2+b^2+c^2+d^2$.
(2) Find all $m$ such that $m=a^3+b^3$, or all $m$ such that $m=a^3+b^3+c^3$, or even $m=a^3+b^3+c^3+d^3$.
(3) Find all $m$ such that $m=p_1+p_2$, or $m=p_1+p_2+p_3$, or even $m=p_1+p_2+p_3+p_4$ for primes $p_i$.
It seems that as you allow for more variables, the more possible $m$ you are able to get. For example, there's Lagrange's four square theorem, the fact every number is the sum of nine positive cubs, and the fact that every sufficiently large number can be written as the sum of four primes.
In reverse, the fewer variables you allow the fewer $m$ you get and the harder to identify which $m$ are attained. There are only so many numbers $m=a^2+b^2$, only so many numbers $m=a^3+b^3$, and certainly $m=p_1+p_2$ excludes most odd numbers and it is unknown if all even integers are of this form.
Waring's Problem states that for each $k$ there exists an $n=g(k)$ such that $m=\sum_{i=1}^m a_i^k$ always has a solution in the natural numbers ($\geq 1$). It seems possible to generalize this idea. My question is as follows:
For some increasing $f:N \to N$, ($N = \{x \in Z | z \geq 1\}$), say with $f(1) = 1$ and $\limsup f(x)/x^p < \infty$ for some $p$, can you show that there exists an $M$ such that if $$G_M:N^M \to N, \qquad G(a_1,\dots, a_M) = \sum_{i=1}^M f(a_i)$$ then $G_M(N^M) = N$? In other words, for any $f$, is there an $M$ such that $$m = f(a_1) + \cdots + f(a_M)$$ has a solution? I am not interested in the minimal $M$, just that one exists. I also know that if $f(x)= 2^{x-1}$, the above fails when considering $m=2^{a-1}$.
It seems to me that the above has not been proven anywhere since the Waring Goldbach Problem seems to be open, but is there a counterexample someone can cook up?
See http://arxiv.org/abs/1401.7598 by Nathanson. there is still nothing easy, but all that is required is that, given an infinite set of positive integers $A,$ if there is some $g$ such that $gA$ has nonzero Shnirel'man density, then there is a larger $h$ such that $hA$ contains all sufficiently large numbers. This is near the top of page 2, just before the paragraph beginning "Landau."
Not clear where he talks about $p$-adic obstructions. If your set $A$ has all even numbers, so does any sum, so you never get odd numbers. Such concerns are in there somewhere, or, at least, in his references.
For limitations such as interest you, evidently Nathanson 1990 and P. Hegedus et al 1998
found a book online called Sumsets and Structure by the same Ruzsa